## Dynamical Systems, Graphs, and AlgorithmsThis book describes a family of algorithms for studying the global structure of systems. By a finite covering of the phase space we construct a directed graph with vertices corresponding to cells of the covering and edges corresponding to admissible transitions. The method is used, among other things, to locate the periodic orbits and the chain recurrent set, to construct the attractors and their basins, to estimate the entropy, and more. |

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### Contents

1 | |

Symbolic Image | 15 |

Periodic Trajectories | 27 |

Newtons Method 35 | 34 |

Invariant Sets | 43 |

Chain Recurrent Set | 55 |

Attractors 65 | 64 |

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### Common terms and phrases

admissible path algorithm approximation bifurcation box M(i cell M(i chain recurrent set chaotic attractor characteristic exponent class H compact complementary differential components computation construction coordinates corresponding covering defined Definition Denote diffeomorphism differential equations discrete domain of attraction dynamical system edge eigenvalues entropy equivalent recurrent vertices exists filtration finite G(PF global attractor graph G Hence homeomorphism homoclinic points hyperbolic fixed point Ikeda map image G intersect invariant curve invariant manifold investigation iterates labeled Let us consider linear Lyapunov exponents mapping f matrix maximal diameter maximal invariant set method Möbius band Morse spectrum non-leaving nonwandering obtain Osipenko p-periodic parameter Per(p periodic orbits periodic points periodic trajectory phase space projective space Proof Proposition pseudo-orbit recurrent cells repellor separatrices set of vertices set Q solution stable and unstable strongly connected components structural graph subdivision symbolic dynamics symbolic image unstable manifold vertex Wu(H